This book is primary intended for solving problems in signals and systems with use of
the MATLABr and thereby understanding the solutions clearly.
This MATLAB based book has resulted from my teaching experience of undergradu-
ate and graduate-level courses of signals and systems, digital image processing over the
past decade.
Today most students prefer watching a screen to taking notes even in class of signals
and systems really needed great e®ort, so it is liable to neglect their studies. Fortunately
most students are familiar with the computers and the internet and also have some expe-
rience of the MATLAB software. The MATLAB is easily accessible to even a beginner,
e®ective in visualization of signals, has an abundance of mathematical and engineering
functions and a great °exibility for user to make new commands and functions for his or
her own purpose.
For the course achievement, it requires as much direct and actual experience as pos-
sible since touching signals and systems for oneself is worth much more than listening,
seeing, or writing down.
For this purpose, we ?rst select a set of 108 topics which are fundamentals in signals
and systems, second introduce and discuss about each selected topic, and at last, set
problems and solve the problems with MATLAB.
This book consists of three chapters. The ?rst chapter introduces the basic mathematics
and 3D visualization techniques with 20 topics such as sinusoid, complex exponential,
sinc function, Dirichlet function, convolution integral, partial fraction expansion, orthog-
onality, 3D mesh plot of 1
s in the complex plane, etc.
The second chapter covers the continuous-time signals and systems with 43 topics such
as Fourier series and transforms, Gibbs phenomenon, generalized impulse trains, sam-
3
4
pling and dual sampling theorems, transfer function and frequency response of a linear
time-invariant (LTI) system, Butterworth lowpass ?lter, 3D mesh plots of the Laplace
transform of the step function and transfer functions of the LTI systems, representation
of Dirac-delta impulse on the rectangular and polar coordinate systems, etc.
The third chapter deals with the discrete-time signals and systems with 45 topics such
as the discrete-time Fourier transform (DTFT), relationship between Dirichlet function
and sinc function, relationship between the DTFT of sgn[n] and the Fourier transform of
sgn(t), the di®erence of u(n) and u[n], periodic convolution, up and down samplings, the
discrete Fourier transform (DFT), the z transform, pole-zero distribution, reconstruc-
tion of the z transform from the samples on jzj = r, analog to digital ?lter conversion,
correction of impulse invariance transformation, digital Butterworth ?lter, 2D and 3D
visualizations of 1D ?ltering, single tone estimation, deconvolution, etc.
In appendix A, some basic and useful formulas derived and used in the book are sum-
marized. Appendix B shows the evaluation of the area under sincN(t) for N = 1; 2 by a
partial area sum technique. In appendix C, the new 14 MATLAB functions developed
for this book are included.
Tae Young Choi
Professor
Division of Electrical and Computer Engineering
College of Information Technology
Ajou University, Korea
Acknowledgements
This work was supported in part by the NEXT (Nurturing EXcellent engineers in in-
formation Technology) project, Division of Electrical and Computer Engineering, Ajou
University. The NEXT is a nationwide project of the Ministry of Information and Com-
munication, Korea.

Preface 1
1 Background 9
1.1 Sinusoidal Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Complex Exponential Waveforms . . . . . . . . . . . . . . . . . . . . . . 12
1.3 3D Line Plot on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Pulse Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Sinc Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Dirichlet Kernel Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Convolution Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Properties of Convolution Integral . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Convolution Integral with Step Signal . . . . . . . . . . . . . . . . . . . . 28
1.10 Convolution between Finite Length Impulse Trains . . . . . . . . . . . . . 31
1.11 Hilbert Transform: Shifting the Phase by ¡90± . . . . . . . . . . . . . . . 33
1.12 Generalized Unit Impulse and Step Functions . . . . . . . . . . . . . . . 36
1.13 Orthogonal Signal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.14 Partial Fraction Expansion of Irrational Polynomials . . . . . . . . . . . . 45
1.15 Partial Fraction Expansion of Laplace and z Transforms . . . . . . . . . . 49
1.16 Area under 1
s on a vertical line in the Complex Plane . . . . . . . . . . . 52
1.17 Contour Integral of 1
s in the Complex Plane . . . . . . . . . . . . . . . . 54
1.18 3D Line Plot of 1
s over the Right Half Complex Plane . . . . . . . . . . . 55
1.19 3D Line Plot of 1
s over the Entire Complex Plane . . . . . . . . . . . . . . 59
1.20 3D Surface Mesh Plot of 1
s close to Re(s) = 0 . . . . . . . . . . . . . . . . 64
2 Continuous-Time Signals and Systems 71
2.1 The Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Fourier Coe±cients of Pulse Train . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Fourier Coe±cients of Recti?ed Waveforms . . . . . . . . . . . . . . . . . 80
2.4 Fourier Coe±cients of Impulse Trains . . . . . . . . . . . . . . . . . . . . 83
2.5 Fourier Coe±cients of Dirichlet Kernel . . . . . . . . . . . . . . . . . . . 86
5
6 CONTENTS
2.6 Fourier Series Expansion of Impulse Train . . . . . . . . . . . . . . . . . . 89
2.7 Fourier Series Expansion of Pulse Train . . . . . . . . . . . . . . . . . . . 92
2.8 Gibbs Phenomenon Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.9 Generalized Impulse Train (GIT) . . . . . . . . . . . . . . . . . . . . . . . 100
2.10 Fourier Series Expansion of Generalized Impulse Trains . . . . . . . . . . 103
2.11 Fourier Series Expansion of Generalized Pulse Train . . . . . . . . . . . . 106
2.12 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.13 Fourier Transform of |(t), sinc(t), and cos(2¼t)|(2t) . . . . . . . . . . . 113
2.14 Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.15 Fourier Transforms of Impulse Trains of Finite Duration . . . . . . . . . 118
2.16 Fourier Transform of Generalized Impulse Functions . . . . . . . . . . . . 123
2.17 Generalized Fourier Transforms of u(t), 1, and sgn(t) . . . . . . . . . . . . 127
2.18 Generalized Fourier Transform of Complex Exponential . . . . . . . . . . 130
2.19 Generalized Fourier Transform of Impulse Train . . . . . . . . . . . . . . 132
2.20 Orthogonality of the sinc function . . . . . . . . . . . . . . . . . . . . . . 135
2.21 Ideal Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.22 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.23 Dual Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
2.24 DC Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2.25 Sampling of |(f) and
P
n sinc(t ¡ nT) . . . . . . . . . . . . . . . . . . . 147
2.26 The Value of a Time-Limited Signal at time origin . . . . . . . . . . . . . 151
2.27 Ideal Lowpass ?lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
2.28 Real Lowpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.29 Lowpass Filtering of Periodic Pulse Train . . . . . . . . . . . . . . . . . . 163
2.30 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
2.31 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
2.32 Single-Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 172
2.33 Frequency Response of a Linear Time-Invariant System . . . . . . . . . . 174
2.34 Analysis of a Second Order LTI System . . . . . . . . . . . . . . . . . . . 177
2.35 Frequency Response of Butterworth Filter . . . . . . . . . . . . . . . . . 181
2.36 Transfer Function of Butterworth Filter . . . . . . . . . . . . . . . . . . . 184
2.37 Filter Conversion from Lowpass Filter . . . . . . . . . . . . . . . . . . . . 186
2.38 Butterworth Filter Conversion . . . . . . . . . . . . . . . . . . . . . . . . 189
2.39 3D Mesh Plot of Laplace Transform of u(t) . . . . . . . . . . . . . . . . . 192
2.40 Generalized Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . 194
2.41 3D Mesh Plot of the Second Order Transfer Functions . . . . . . . . . . . 200
2.42 The Limiting Function of 1¡e¡j!T
j! as T ! 1 . . . . . . . . . . . . . . . . 204
2.43 Dirac-Delta Impulse on the Complex Plane . . . . . . . . . . . . . . . . . 207
CONTENTS 7
3 Discrete-Time Signals and Systems 217
3.1 The Discrete-Time Fourier Transform (DTFT) . . . . . . . . . . . . . . . 219
3.2 DTFT of Windowed Complex Exponential . . . . . . . . . . . . . . . . . . 224
3.3 DTFT of Windowed Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . 230
3.4 DTFT of Rectangular Pulse and dc Sequences . . . . . . . . . . . . . . . 235
3.5 DTFT of Triangular Pulse Sequence . . . . . . . . . . . . . . . . . . . . . 239
3.6 Dirichlet and sinc functions . . . . . . . . . . . . . . . . . . . . . . . . . . 243
3.7 DTFTs of sgn[n] Based on the FT of sgn(t) . . . . . . . . . . . . . . . . . 245
3.8 DTFT of Exponential Sequence from the FT of Exponential Signal . . . . 248
3.9 Characteristics of Q(r; !) = 1+re¡j!
1¡re¡j! . . . . . . . . . . . . . . . . . . . . . 252
3.10 Sum of an In?nite Geometric Series . . . . . . . . . . . . . . . . . . . . . . 256
3.11 DTFTs of u(n); u[n], sgn[n], and 1 . . . . . . . . . . . . . . . . . . . . . . 261
3.12 DTFT of a Causal Rectangular Pulse Sequence . . . . . . . . . . . . . . . 264
3.13 Mesh Plot of the DTFT of rnu[n] . . . . . . . . . . . . . . . . . . . . . . . 268
3.14 Digital Di®erentiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
3.15 Periodic Convolution in Frequency Domain . . . . . . . . . . . . . . . . . 274
3.16 Upsampling and Downsampling . . . . . . . . . . . . . . . . . . . . . . . . 277
3.17 Samplings in Both Time and Frequency Domains (I) . . . . . . . . . . . . 285
3.18 Sampling in Both Time and Frequency Domains (II) . . . . . . . . . . . . 292
3.19 The Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . 297
3.20 Relationship between the DFT and the DTFT . . . . . . . . . . . . . . . 301
3.21 Circular Shift in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
3.22 DTFT and DFT of Complex Exponential Sequence . . . . . . . . . . . . 310
3.23 Periodic Convolution via Circular Convolution . . . . . . . . . . . . . . . 314
3.24 The z Transform and Its Properties . . . . . . . . . . . . . . . . . . . . . 319
3.25 z Transform of a Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . 324
3.26 z Transform of rjnj for jnj · N . . . . . . . . . . . . . . . . . . . . . . . . 328
3.27 Generalized z Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
3.28 Sampling the z Transform on the Unit Circle . . . . . . . . . . . . . . . . 342
3.29 Reconstruction of the z Transform on jzj = 1 . . . . . . . . . . . . . . . . 348
3.30 Reconstruction of the z Transform on jzj = r . . . . . . . . . . . . . . . . 352
3.31 Frequency Responses of Exponentials . . . . . . . . . . . . . . . . . . . . 359
3.32 Impulse Invariance Transformation (IIT) . . . . . . . . . . . . . . . . . . . 364
3.33 Digital Elliptic Filter Design Using the IIT . . . . . . . . . . . . . . . . . 371
3.34 Discrete-Time Linear Time-Invariant (DLTI) System . . . . . . . . . . . . 374
3.35 DC Blocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
3.36 Analysis of a Di®erence Equation y[n] + r2y[n ¡ 2] = x[n] . . . . . . . . . 383
3.37 DLTI System of the Second Order Lowpass Filter . . . . . . . . . . . . . . 389
3.38 DLTI System of the Second Order Highpass Filter . . . . . . . . . . . . . 393
8 CONTENTS
3.39 Bilinear Transformation (BT) and Frequency Warping . . . . . . . . . . . 396
3.40 Digital Filter Designs Using the IIT and the BT . . . . . . . . . . . . . . 401
3.41 Pole and Zero Distributions of Digital Filters . . . . . . . . . . . . . . . . 405
3.42 2D and 3D Visualizations of 1D Digital Filter (I) . . . . . . . . . . . . . . 409
3.43 2D and 3D Visualizations of 1D Digital Filter (II) . . . . . . . . . . . . . 413
3.44 Single Tone Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
3.45 Deconvolution (Inverse Filtering) . . . . . . . . . . . . . . . . . . . . . . . 423
Appendix 427
A Mathematical Formulas 427
B Evaluation of the area under sincN(t) 431
C Matlab Functions 433
Index 450